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Theorem csbriotag 6333
Description: Interchange class substitution and restricted description binder. (Contributed by NM, 24-Feb-2013.)
Assertion
Ref Expression
csbriotag  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( iota_ y  e.  B ph )  =  ( iota_ y  e.  B [. A  /  x ]. ph ) )
Distinct variable groups:    y, A    x, B    x, y
Allowed substitution hints:    ph( x, y)    A( x)    B( y)    V( x, y)

Proof of Theorem csbriotag
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3097 . . 3  |-  ( z  =  A  ->  [_ z  /  x ]_ ( iota_ y  e.  B ph )  =  [_ A  /  x ]_ ( iota_ y  e.  B ph ) )
2 dfsbcq2 3007 . . . 4  |-  ( z  =  A  ->  ( [ z  /  x ] ph  <->  [. A  /  x ]. ph ) )
32riotabidv 6322 . . 3  |-  ( z  =  A  ->  ( iota_ y  e.  B [
z  /  x ] ph )  =  ( iota_ y  e.  B [. A  /  x ]. ph )
)
41, 3eqeq12d 2310 . 2  |-  ( z  =  A  ->  ( [_ z  /  x ]_ ( iota_ y  e.  B ph )  =  ( iota_ y  e.  B [
z  /  x ] ph )  <->  [_ A  /  x ]_ ( iota_ y  e.  B ph )  =  ( iota_ y  e.  B [. A  /  x ]. ph )
) )
5 vex 2804 . . 3  |-  z  e. 
_V
6 nfs1v 2058 . . . 4  |-  F/ x [ z  /  x ] ph
7 nfcv 2432 . . . 4  |-  F/_ x B
86, 7nfriota 6330 . . 3  |-  F/_ x
( iota_ y  e.  B [ z  /  x ] ph )
9 sbequ12 1872 . . . 4  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
109riotabidv 6322 . . 3  |-  ( x  =  z  ->  ( iota_ y  e.  B ph )  =  ( iota_ y  e.  B [ z  /  x ] ph ) )
115, 8, 10csbief 3135 . 2  |-  [_ z  /  x ]_ ( iota_ y  e.  B ph )  =  ( iota_ y  e.  B [ z  /  x ] ph )
124, 11vtoclg 2856 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( iota_ y  e.  B ph )  =  ( iota_ y  e.  B [. A  /  x ]. ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632   [wsb 1638    e. wcel 1696   [.wsbc 3004   [_csb 3094   iota_crio 6313
This theorem is referenced by:  cdlemkid3N  31744  cdlemkid4  31745
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-riota 6320
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